Problem 5

Question

Find the derivative of the following polynomials in $\mathbb{Z}_{5}[x]$ : $$ x^{6}+2 x^{3}+x+1 \quad x^{5}+3 x^{2}+1 \quad x^{15}+3 x^{10}+4 x^{5}+1 $$

Step-by-Step Solution

Verified

Answer

The derivatives of the polynomials are: 1) $x^5 + x^2 + 1$, 2) $x$, 3) $0$.

1Step 1: Identify the rules of differentiation in polynomial

The derivative of a polynomial function is obtained by applying the power rule, which states that the derivative of $ ax^n $ is $ n imes ax^{n-1} $. This rule holds for each term in the polynomial individually.

2Step 2: Differentiate each polynomial separately

We process each term in the polynomial according to the power rule and then apply the modulo operation as required.For the polynomial $ x^6 + 2x^3 + x + 1 $: - $ \frac{d}{dx}(x^6) = 6x^5 \equiv 1x^5 \mod 5 $ - $ \frac{d}{dx}(2x^3) = 6x^2 \equiv 1x^2 \mod 5 $ - $ \frac{d}{dx}(x) = 1 $ - Constant term derivative is 0.Thus, the derivative is $ x^5 + x^2 + 1 $.For the polynomial $ x^5 + 3x^2 + 1 $:- $ \frac{d}{dx}(x^5) = 5x^4 \equiv 0 \mod 5 $- $ \frac{d}{dx}(3x^2) = 6x \equiv 1x \mod 5 $- Constant term derivative is 0.Thus, the derivative is $ x $.For the polynomial $ x^{15} + 3x^{10} + 4x^5 + 1 $:- $ \frac{d}{dx}(x^{15}) = 15x^{14} \equiv 0 \mod 5 $- $ \frac{d}{dx}(3x^{10}) = 30x^9 \equiv 0 \mod 5 $- $ \frac{d}{dx}(4x^5) = 20x^4 \equiv 0 \mod 5 $Thus, the derivative is $ 0 $.

3Step 3: Conclusion of results

We have simplified the derivatives by applying the modulo operation. For each polynomial, the derivatives are:1. $ x^6 + 2x^3 + x + 1 $ results in $ x^5 + x^2 + 1 $.2. $ x^5 + 3x^2 + 1 $ results in $ x $.3. $ x^{15} + 3x^{10} + 4x^5 + 1 $ results in $ 0 $. These expressions are the derivatives in the ring $ \mathbb{Z}_5[x] $.

Key Concepts

Power Rule in Polynomial DifferentiationModulo Operation in DifferentiationUnderstanding Ring $\mathbb{Z}_5[x]$

Power Rule in Polynomial Differentiation

In calculus, the power rule is a fundamental tool for differentiating polynomial functions. It provides a straightforward method to find the derivative of terms in the form of $ ax^n $, where $ a $ is a constant and $ n $ is a non-zero integer. The power rule states that to differentiate $ ax^n $, you multiply the term by the exponent $ n $ and decrease the exponent by one, yielding $ n \times ax^{n-1} $.
This rule is applied to each term independently within a polynomial. For example, in the polynomial $ x^6 + 2x^3 + x + 1 $, each term is differentiated separately:

For $ x^6 $, applying the power rule gives $ 6x^5 $.
For $ 2x^3 $, it becomes $ 6x^2 $.
$ x $ simplifies to 1, as any constant term like 1 results in a derivative of 0.

Thus, understanding the power rule is crucial as it forms the basis for polynomial differentiation in any context, including modular arithmetic.

Modulo Operation in Differentiation

The modulo operation is key when working with polynomial expressions within modular arithmetic. When we differentiate polynomials and apply the power rule, we might end up with coefficients that are larger than the modulus we’re working with. In our case, we are working modulo 5, represented as $ \mathbb{Z}_5[x] $.
As part of this operation, each coefficient derived from differentiation is reduced to its equivalent value within the set from 0 to 4 by using the modulo operation. This means replacing any coefficient $ c $ with $ c \mod 5 $.
Consider the derivative $ 6x^5 $ from the polynomial $ x^6 $:

The coefficient 6 becomes $ 6 \equiv 1 \mod 5 $, which reduces the term to $ 1x^5 $.

Similarly, for $ 6x^2 $, 6 modulo 5 is also 1, simplifying the term to $ x^2 $.
Therefore, the modulo operation ensures that the results remain within the bounds of our defined ring, maintaining consistency and accuracy in the solution.

Understanding Ring $\mathbb{Z}_5[x]$

A ring like $\mathbb{Z}_5[x]$ is a mathematical set equipped with two operations that resemble addition and multiplication from algebra. Here, $ \mathbb{Z}_5[x] $ indicates polynomials with coefficients that are integers from the set $ \{0, 1, 2, 3, 4\} $.
Working within this ring means all operations on these polynomials are calculated modulo 5. Here's why that matters:

It brings a cyclic nature to the arithmetic. For example, a coefficient of 5 is equivalent to 0, simplifying mathematical expressions greatly.
This approach is particularly useful in fields like cryptography, where working with limited sets of numbers enhances security.

Operating in a ring like $\mathbb{Z}_5[x]$ helps in ensuring consistency in results by keeping all polynomial terms within a defined boundary. This makes it easier to handle complex polynomial computations in a manageable framework.

Problem 4

Problem 5